Download A new approach for the two-electron cumulant in natural orbital

A New Approach for the Two-Electron
Cumulant in Natural Orbital
Functional Theory
MARIO PIRIS
Institute of Physical and Theoretical Chemistry, Friedrich-Alexander University
Erlangen-Nuremberg, Egerlandstrasse 3, 91058 Erlangen, Germany
Received 11 May 2005; accepted 29 August 2005
Published online 4 November 2005 in Wiley InterScience (www.interscience.wiley.com).
DOI 10.1002/qua.20858
ABSTRACT: The cumulant expansion gives rise to a useful decomposition of the
two-matrix D in which the pair correlated matrix (cumulant) is disconnected from the
antisymmetric product of the one-matrix ⌫. A new explicit antisymmetric approach for
the two-particle cumulant matrix in terms of two symmetric matrices, ⌬ and ⌳, as
functionals of the occupation numbers is proposed for singlet ground states of closedshell systems. It produces a natural orbital functional that reduces to the exact
expression for the total energy in two-electron systems. The functional form of matrix ⌳
is readily generalized to any system with an even number of electrons. The diagonal
elements of ⌬ equal the square of the occupation numbers, and the N-representability
positivity necessary conditions of the two-matrix impose several bounds on the offdiagonal elements of matrix ⌬. The well-known mean value theorem and the partial
sum rule obtained for the off-diagonal elements of ⌬ provide a prescription for deriving
a practical functional. In particular, when the mean values { J i*} of the Coulomb
interactions { Jij} for a given orbital i taking over all orbitals j ⫽ i are assumed to be
equal {Kii/2}, a functional close to self-interaction-corrected GU functional is obtained,
but the two-matrix fermionic antisymmetric holds. An additional term for the matrix
elements of ⌳ between HF occupied orbitals is proposed to ensure a correct description
of the occupation numbers for the lowest occupied levels. The functional is tested in
fully variational finite basis set calculations of 57 molecules. It gives reasonable
molecular energies at the equilibrium geometries. The calculated values of dipole
moments are in good agreement with the available experimental data. © 2005 Wiley
Periodicals, Inc. Int J Quantum Chem 106: 1093–1104, 2006
Key words: natural orbital functional; cumulant; reduced density matrix; dipole
moment; electron correlation energies
Contract grant sponsor: Alexander von Humboldt (AvH)
Foundation.
International Journal of Quantum Chemistry, Vol 106, 1093–1104 (2006)
© 2005 Wiley Periodicals, Inc.
PIRIS
1. Introduction
T
he idea of a one-particle reduced density matrix (one-matrix) ⌫ functional appeared some
decades ago [1]. More recently, several functionals
of the natural orbitals and their occupation numbers have been proposed [2–11]. A major advantage
of the method is that the kinetic energy and the
exchange energy are explicitly defined using the
one-matrix and do not require the construction of a
functional. The unknown functional only needs to
incorporate electron correlation. Moreover, the onematrix is a much simpler object than the N-particle
wave function, and the ensemble N-representability
conditions that have to be imposed on variations of
⌫ are well known [12]. Finally, the natural orbital
functional (NOF) incorporates fractional occupation numbers in a natural way, which provides a
correct description of both dynamical and nondynamical correlation.
An NOF requires an expression of the two-particle reduced density matrix (two-matrix) D in
terms of the one-matrix ⌫. Such reconstruction of
the two-matrix can be achieved using the wellknown cumulant expansion of D [13]. In the
present study, we propose an explicit antisymmetric form for the two-particle cumulant matrix. By
employing an antisymmetric ansatz for the twomatrix, we can obtain a correct description of the
pair density for parallel-spin electrons, which is
poorly described by its predecessor [11].
Unlike density functional theory (DFT), density
matrix functional theory affords an exact energy
functional for two-electron systems [6, 14]. The ansatz presented here can be reduced to this exact
expression providing the specific form of the cumulant matrix in the two-electron case. This functional
form can be readily generalized to any N-electron
system, but the off-diagonal elements of a symmetric matrix ⌬ remain unknown. In contrast, several
constraints for these magnitudes can be achieved
via some known positivity conditions. In the
present work, we analyze the D, G, and Q conditions of two-matrix N-representability, and establish several bounds on the off-diagonal elements of
matrix ⌬.
To develop a practical method, a further approximation is assumed. Instead of the approach of
off-diagonal elements of ⌬ considering constraints
imposed by positivity conditions, we approximate
the term that involves matrix ⌬, by employing the
well-known mean value theorem and the partial
1094
sum rule obtained for this matrix. This leads to a
total energy close to the self-interaction-corrected
Hartree functional proposed by Goedecker and
Umrigar (GU) [6]. The GU functional is not antisymmetric assuming a Hartree-product form for
the opposite spin component of D, so it does not
afford an N-representable two-matrix [15]. In contrast to this reconstruction, the current proposal can
satisfy not only the hermiticity and particle permutation conditions, but also the trace relation and
positivity conditions of the two-matrix. Besides, it
appears to reproduce properly the occupation numbers for lower occupied levels.
We begin with a presentation of the basic concepts and notations relevant to NOF theory (Section
2). We then present our ansatz for the two-electron
cumulant matrix (Section 3). The two- and N-electron cases, as well as the N-representability of the
functional, are discussed in detail. The following
section is devoted to our further simplification to
achieve a practical functional (Section 4). We end
with a short presentation of the methodological
details (Section 5) and some results for selected
molecules (Section 6).
2. Basic Concepts and Notations
2.1. GENERAL FORMALISM
We consider an N-electron Coulombic system
described by the Hamiltonian
Ĥ ⫽
冘 h ⌫ˆ ⫹ 冘 具ij兩kl典D̂
ij
ji
ij
kl,ij
,
(1)
ijkl
where hij denotes the one-electron matrix elements
of the core-Hamiltonian,
h ij ⫽
冕冕
冋
1
dx ␹i*共x兲 ⫺ ⵜ2 ⫺
2
冘 兩r ⫺Z r 兩册␹ 共x兲,
I
I
I
j
(2)
and 具ij兩kl典 denotes the two-electron matrix elements
of the Coulomb interaction
具ij兩kl典 ⫽
冕冕
⫺1
dx 1dx2␹i*共x1兲␹j*共x2兲r12
␹k共x1兲␹l共x2兲.
(3)
Atomic units are used. Here and in the following
x ⬅ (r, s) stands for the combined spatial and spin
coordinates, r and s, respectively. The spin-orbitals
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TWO-ELECTRON CUMULANT IN NATURAL ORBITAL FUNCTIONAL THEORY
{␹i(x)} constitute a complete orthonormal set of single-particle functions,
具 ␹ i兩 ␹ j典 ⫽
冕
dx ␹i*共x兲␹j共x兲 ⫽ ␦ij,
⌫ˆ ji ⫽ â †j â i
(5)
冉冊
(6)
and
1 † †
â â â â
2 k l j i
are constructed from the familiar creation and annihilation operators, {â†i } and {âi} [16], respectively,
associated with the set of spin-orbitals {␹i(x)}.
A quantum mechanical pure state of our N-particle system can be characterized by a normalized
wave function ⌿ or a system density matrix ⌫N
⌫ N 共x⬘1, x⬘2, . . . , x⬘N; x1, x2, . . . , xN兲
⫽ ⌿*共x⬘1, x⬘2, . . . , x⬘N兲⌿共x1, x2, . . . , xN兲.
Tr D ⫽
(4)
with the usual meaning of the Kronecker delta ␦ij.
The one- and two-particle density matrix operators,
D̂ kl,ij ⫽
and the trace of the two-matrix gives the number of
electron pairs in the system
(7)
冘 h ⌫ ⫹ 冘 具ij兩kl典D
ij
ji
ij
kl,ij
,
(8)
ijkl
where the one- and two-particle reduced density
matrices, or briefly the one- and two-matrices, are
defined as
⌫ ji ⫽ 具⌿兩⌫ˆ ji 兩⌿典
(9)
D kl,ij ⫽ 具⌿兩D̂ kl,ij 兩⌿典.
(10)
ij,ij
ij
Tr ⌫ ⫽
冘 ⌫ ⫽ N,
ii
i
(11)
冉冊
N共N ⫺ 1兲
N
⫽ 2 .
2
D kl,ij ⫽ D *ij,kl
(12)
(13)
and the antisymmetry,
D kl,ij ⫽ ⫺Dlk,ij ⫽ ⫺Dkl, ji ⫽ Dlk, ji.
(14)
An important contraction relation between oneand two-matrices is in agreement with the previous
normalization [Eqs. (11) and (12)]:
⌫ ji ⫽
2
N⫺1
冘D
jk,ik
.
(15)
k
This implies that the energy functional (8) refers
only to the two-matrix, because D determines ⌫.
Attempts to determine the energy by minimizing E
[D] are complicated because of the lack of a simple
set of necessary and sufficient conditions for ensuring that the two-matrix corresponds to an N-particle wave function (the N-representability problem)
[12]. Alternatively, the last term in Eq. (8) can be
replaced by an unknown functional of the onematrix [18]:
E关⌫兴 ⫽
According to expression (8), the energy E of a state
⌿ is an exactly and explicitly known functional of ⌫
and D.
The reduced density matrices satisfy important
sum rules [17]. The trace (Tr) of the one-matrix
equals the number of electrons,
⫽
Their diagonal elements in the coordinate-space
representation are always non-negative, since ⌫(x1;
x1) is related to the probability of finding one electron at x1, and D(x1, x2; x1, x2) is related to the
probability of finding one electron at x1 and another
at x2.
D satisfies several relations that follow directly
from the anticommutation rules and the hermiticity
of the operators {â†i } and {âi}, namely, the hermiticity,
The expectation value of the Hamiltonian (1) for
the state ⌿ is then
E⫽
冘D
冘 h ⌫ ⫹ V 关⌫兴.
ij
ji
ee
(16)
ij
The functional Vee[⌫] is universal in the sense
that it is independent of the external field. Its properties are well known [19]. However, it is very
difficult to approximate because what we have
done is to change the variational unknown from the
complicated many-variable function ⌿ to a single
one-matrix ⌫.
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PIRIS
The one-matrix ⌫ can be diagonalized by a unitary transformation of the spin-orbitals {␹i(x)}, with
the eigenvectors being the natural spin-orbitals and
the eigenvalues {ni} representing the occupation
numbers of the latter,
⌫ ji ⫽ n i ␦ ji .
(17)
In the following, all representations used are
assumed to refer to this basis. Accordingly, the
energy functional is determined by the natural orbitals and their occupation numbers, i.e., E[{ni, ␺i}].
In addition, we assume all orbitals to be real.
冘D
␣␤
jk,ik
⫽
k
N ␣␣ N
⌫ ⫽ n i ␦ ji .
4 ji
4
(23)
It is readily demonstrated that the sum rules (22)
and (23) are compatible with Eq. (15). The traces of
these two-matrix components read
Tr D ␣␣ ⫽
N共N ⫺ 2兲
,
8
Tr D␣␤ ⫽
N2
.
8
(24)
Only singlet spin-compensated systems will be considered in the remainder of this study.
2.2. RESTRICTED FORMALISM
The Hamiltonian (1) is spin independent, so only
density matrix blocks that conserve the number of
each spin type are nonvanishing. Specifically, the
one-matrix has two nonzero blocks, an ␣ block (⌫␣)
and a ␤ block (⌫␤),
⌫ ␣␣
ji ⫽ 0,
⌫ ␤␤
ji ⫽ 0,
(18)
and the two-matrix has three nonzero blocks, an ␣␣
block (D␣␣), an ␣␤ block (D␣␤), and a ␤␤ block
(D␤␤),
␣␣ , ␣␣
⫽ 0,
D kl,ij
␣␤ , ␣␤
D kl,ij
⫽ 0,
␤␤ , ␤␤
D kl,ij
⫽ 0.
␺ ␣i 共r兲 ⫽ ␺␤i 共r兲 ⫽ ␺i共r兲.
3.1. CUMULANT OF THE TWO-MATRIX
It remains to find approximations for the unknown functional Vee[⌫]. To this end, we use a
reconstructive functional D [⌫]; that is, we express
elements Dkl,ij in terms of ⌫ji. We neglect any explicit dependence of D on the natural orbitals themselves because the energy functional (8) already has
a strong dependence on the natural orbitals via the
one- and two-electron integrals. Let us consider the
following well-known [13] decomposition of D:
(19)
The set of spin-orbitals {␹i(x)} is split here into
two subsets: {␺␣i (r)␣(s)} and {␺␤i (r)␤(s)}. In the case
of spin compensated systems, the two blocks of the
one-matrix are the same (⌫␣ ⫽ ⌫␤), i.e.,
n ␣i ⫽ n ␤i ⫽ n i ,
3. Two-Matrix
(20)
␣␣
D kl,ij
⫽
1 ␣ ␣
␣␣
共⌫ ⌫ ⫺ ⌫ kj␣ ⌫ ␣li ⫹ ␭ kl,ij
兲
2 ki lj
n jn i
␣␣
共 ␦ ki ␦ lj ⫺ ␦ kj ␦ li 兲 ⫹ ␭ kl,ij
2
(25)
1 ␣ ␤
1
␣␤
␣␤
共⌫ ⌫ ⫹ ␭ kl,ij
兲 ⫽ 共n j n i ␦ ki ␦ lj ⫹ ␭ kl,ij
兲,
2 ki lj
2
(26)
⫽
␣␤
⫽
D kl,ij
The trace of the one-matrix (11) becomes
2
冘 n ⫽ N.
i
(21)
i
For singlet states, the first and the last blocks of
the two-matrix are also equal (D␣␣ ⫽ D␤␤), so in
this work we deal only with D␣␣ and D␣␤. The
parallel-spin component D␣␣ is antisymmetric, but
D␣␤ possess no special symmetry. Each of these
two-matrix blocks must contract to the appropriate
one-matrix block:
冘D
k
1096
␣␣
jk,ik
⫽
共N ⫺ 2兲 ␣␣ 共N ⫺ 2兲
⌫ ji ⫽
n i ␦ ji
4
4
(22)
where ␭ is the cumulant matrix. It should be noted
that matrix elements of ␭ are nonvanishing only if
all its labels refer to partially occupied natural orbitals with an occupation number different from 0
or 1 [13]. Taking into account the normalization
condition for the one-matrix (21), it can easily be
shown from Eqs. (22) and (23) that spin components
of ␭ fulfill the following sum rules:
冘␭
␣␣
jk,ik
⫽ n i 共n i ⫺ 1兲 ␦ ji
(27)
k
冘␭
␣␤
jk,ik
⫽ 0.
(28)
k
VOL. 106, NO. 5
TWO-ELECTRON CUMULANT IN NATURAL ORBITAL FUNCTIONAL THEORY
The first two terms on the right-hand side of Eq.
(25) together satisfy property (14) of D␣␣. Therefore,
the matrix ␭␣␣ should be antisymmetric as well. In
general, the cumulant has a dependence of four
indices, and it is not feasible to apply direct computation with such quantities to large systems. In
this work, we consider the following parallel component of the cumulant matrix:
␭
␣␣
kl,ij
⌬ji
⌬ji
⫽⫺
␦ki␦lj ⫹ ␦kj␦li,
2
2
(29)
where ⌬ is a symmetric matrix. The sum rule (27)
and the approximate ansatz (29) imply the constraint
冘⬘ ⌬ ⫽ n 共1 ⫺ n 兲.
ji
i
Taking into account that Lij ⫽ Kij for real orbitals,
the expression (34) can be rewritten as
E⫽2
冘 n h ⫹ 2 冘 共n n ⫺ ⌬ 兲J ⫺ 冘 ⌳ K .
i ii
j
i
i
ji
(35)
NOF theory provides an exact energy functional
for two-electron systems [6, 14]. In the weak correlation limit, the total energy is given by
冘 nh ⫹n K
⬁
E⫽2
i ii
1
冘 冑n n K
⬁
11 ⫺ 2
i⫽1
1
i
i1
i⫽2
冘 冑n n K .
⬁
⫹
j
i
ij
(36)
i, j⫽2
The prime indicates that the i ⫽ j term is omitted.
For ␭␣␤, we can achieve a suitable approximation if
we replace the second term in Eq. (29) with the
dependence obtained in the improved Bardeen–
Cooper–Schrieffer (IBCS) method [20], i.e.,
␣␤
␭ kl,ij
⫽⫺
⌬ji
⌸ki
␦ ␦ ⫹
␦ ␦,
2 ki lj
2 kl ij
(31)
where we have introduced a new symmetric matrix
⌸. For purposes of convenience, as we see below,
we define the matrix ⌸ in terms of a new symmetric
matrix ⌳:
⌸ ki ⫽ n k n i ⫺ ⌬ ki ⫺ ⌳ ki .
(32)
Combining Eqs. (28), (30), (31), and (32) results in
2⌬ ii ⫹ ⌳ ii ⫽ 2n 2i ⫺ n i .
冘 n h ⫹ 冘 共n n ⫺ ⌬ 兲共2J ⫺ K 兲
⫹ 冘 共n n ⫺ ⌬ ⫺ ⌳ 兲 L ,
i ii
j
i
ji
ij
ij
ij
j
i
ji
As can be seen from Eq. (36), the dependence of D
on the natural occupations requires a distinction
between HF occupied and virtual orbitals.
Since D␣␣ ⫽ 0 for N ⫽ 2, one easily deduces from
Eqs. (25) and (29) that ⌬ji ⫽ njni. Consequently, it is
not difficult to see from Eqs. (26), (31), and (32) that
D␣␤ nonzero elements have the form D␣␤
jj,ii ⫽
⫺⌳ji/2. Thus, the total energy (35) turns into
E⫽2
ji
ij
(34)
ij
with Jij ⫽ 具ij兩ij典, Kij ⫽ 具ij兩ji典, and Lij ⫽ 具ii兩jj典 [see Eq. (3)
with ␹i(x) replaced by ␺i(r)]. Note that if ⌬ji ⫽ 0 and
⌳ji ⫽ njni (so ⌸ji ⫽ 0), the reconstruction proposed
here yields the Hartree–Fock (HF) case, as expected.
冘 nh ⫺ 冘 ⌳ K .
i ii
i
ji
(37)
ij
ij
From the requirement that for any two-electron
system the expression (37) should yield Eq. (36),
one has to set
⌳ ji ⫽ sign共⌳ji兲 冑njni,
再
⫺1
⫽ ⫹1
⫺1
(33)
Using Eqs. (17), (20), (25), (26), (29), (31), and (32),
the energy in Eq. (8) reads as
i
ij
3.2. TWO-ELECTRON SYSTEMS
j
E⫽2
ji
ij
(30)
i
ij
ij
sign共⌳ji兲
if i ⫽ 1, j ⫽ 1
if i ⫽ 1, j ⱖ 2;
if i ⱖ 2, j ⱖ 2
冎
i ⱖ 2, j ⫽ 1
(38)
It is worth noting that the chosen ⌬ and ⌳ satisfy
constraints (30) and (33). Moreover, the sign rule
(38) also holds true for systems in which the largest
occupation deviates significantly from one, indicating that it may possibly be valid for arbitrary correlation strengths [6].
3.3. N-ELECTRON SYSTEMS
Expression (38) suggests the following functional
form for {⌳ji}:
INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY
1097
PIRIS
⌳ ii ⫽ ⫺ni
(39)
⌳ ji ⫽ sign共⌳ji兲 冑njni
i ⫽ j,
if
(40)
The so-called D, G, and Q conditions state that
the two-electron density matrix, Eq. (10), the electron-hole density matrix G,
G kl,ij ⫽
where
sign共⌳ji兲
再
⫹1
⫽ ⫹1
⫺1
if i ⱕ nco, j ⱕ nco
if i ⱕ nco, j ⬎ nco;
if i ⬎ nco, j ⬎ nco
冎
i ⬎ nco,
1
具⌿兩â k† â l â †j â i 兩⌿典,
2
(44)
and the two-hole density matrix Q,
j ⱕ nco ,
Q kl,ij ⫽
1
具⌿兩â k â l â †j â †i 兩⌿典,
2
(45)
(41)
where the number of HF closed shells is denoted
nco. Inserting Eq. (39) into the equality (33), affords
⌬ ii ⫽ n 2i .
(42)
By taking into account Eqs. (39) and (42), the
energy functional (35) can be expressed as
E⫽
冘 n 共2h ⫹ K 兲 ⫹ 2 冘⬘ 共n n ⫺ ⌬ 兲J
⫺ 冘⬘ ⌳ K .
i
i
ii
ii
j
i
ji
must be positive semidefinite. There are also other
two necessary conditions for the N-representability
of a fermion two-matrix, which are called B and C
conditions. A discussion of relations among these
positivity conditions is given in Ref. [22].
Using the anticommutation relations for the creation and annihilation operators and definitions of
the one- and two-matrices given in Eqs. (9) and (10),
the spin component matrices of G and Q can be
derived from the matrices D and ⌫ as follows:
ij
ij
ji
ij
(43)
␣␣
G kl,ij
⫽
1
␣␣
␦ ⌫ ␣ ⫺ D kj,il
2 lj ki
(46)
␣␤
G kl,ij
⫽
1
␣␤
␦ ⌫ ␣ ⫺ D kj,il
2 lj ki
(47)
ij
Unfortunately, ⌬ji ⫽ njni, taken from the N ⫽ 2
case, violates the sum rule (30) in the general case of
N ⬎ 2. This means that the functional form of
nondiagonal elements of ⌬ remains unknown for
N-electron systems. Nevertheless, some constraints
can be achieved for these quantities, using known
necessary conditions of two-matrix N-representability.
3.4. N-REPRESENTABILITY
The one-matrix and the functional N-representability problems are completely different. Restriction of the occupation numbers {ni} to the range 0 ⱕ
ni ⱕ 1 represents a necessary and sufficient condition for ensemble N-representability of the onematrix [12]. However, the functional N-representability refers to the conditions that guarantee the
one-to-one correspondence between E[⌿] and E[⌫],
a problem related to the N-representability of the
two-matrix. Therefore, any approximation for
Vee[{ni, ␺i}] must comply at least with the known
necessary conditions for the N-representability of
the two-matrix [21].
1098
␣␣
Q kl,ij
⫽
1
共 ␦ ␦ ⫺ ␦ ki ⌫ ␣jl ⫺ ␦ lj ⌫ ␣ik ⫺ ␦ kj ␦ li ⫹ ␦ kj ⌫ ␣il
2 ki lj
␣␤
Q kl,ij
⫽
␣␣
⫹ ␦ li ⌫ jk␣ 兲 ⫹ D ij,kl
(48)
1
␣␤
共 ␦ ␦ ⫺ ␦ ki ⌫ ␤jl ⫺ ␦ lj ⌫ ␣ik 兲 ⫹ D ij,kl
.
2 ki lj
(49)
A matrix is positive semidefinite if, and only if,
all its eigenvalues are non-negative. The solution of
the eigenproblem for D␣␣ is readily carried out,
yielding the following set of eigenvalues:
d ␣␣ ⫽ 兵0,
n j n i ⫺ ⌬ ji 其,
j ⫽ i.
(50)
For more details, the reader can find an analogous derivation in Ref. [15]. D␣␤ consists of 1 ⫻ 1
blocks, along with a single R ⫻ R block, where R is
the number of orbitals. The latter has elements
D ␣␤
ii, jj ⫽
1
共n n ⫺ ⌬ ji ⫺ ⌳ ji 兲.
2 j i
(51)
VOL. 106, NO. 5
TWO-ELECTRON CUMULANT IN NATURAL ORBITAL FUNCTIONAL THEORY
The 1 ⫻ 1 blocks have elements D␣␤
ij,ij, for j ⫽ i,
and thus yield eigenvalues
1
d ␣␤
共n n ⫺ ⌬ ji 兲.
ji ⫽
2 j i
再
⍀ ji ⫺ ⌬ ji ,
⫾
(52)
To sum up, we have analytic expressions for all
eigenvalues of D, except those arising from the
single R ⫻ R block. Consequently, our reconstructive functional satisfies the D condition (d ⱖ 0) if
⌬ji ⱕ njni and the R ⫻ R block of D␣␤ is positive.
Introducing the abbreviation ⍀ji ⫽ 1 ⫺ nj ⫺ ni ⫹
njni, and considering that Q has the same block
structure as D, one has the following set of analytic
eigenvalues:
q ⫽ 0,
n j ⫹ n i ⌬ ji ⫺ n j n i
⫹
4
2
g ␣␤
ji ⫽
冎
⍀ ji ⫺ ⌬ ji
,
2
j ⫽ i.
(53)
Accordingly, if one takes ⌬ji ⱕ ⍀ji, and the R ⫻ R
block of Q␣␤ is positive, the Q condition is fulfilled.
It is easy to verify that the D condition is more
restrictive than the Q condition for ⌬ji between HF
virtual orbitals (occupation numbers are close to
zero), whereas for elements between HF occupied
levels, the Q condition is predominant.
Finally, we consider G. The spin component G␣␣
also contains a single block R ⫻ R, for which the
eigenvalues have no analytic expression, and 1 ⫻ 1
blocks. These latter blocks contribute eigenvalues
1
4
冑 共n j ⫺ n i兲 2 ⫹ 4共n jn i ⫺ ⌬ ji ⫺ ⌳ ji兲 2.
To ensure that g␣␤
ji ⱖ 0, the expression (56) gives
rise to the inequality
⌬ ji ⱕ n j n i ⫹
n j n i ⫺ ⌳ 2ji
.
2⌳ ji ⫺ n j ⫺ n i
To obtain the NOF for a system of N electrons,
one may attempt to approximate off-diagonal elements of ⌬ considering the sum rule (30), as well as
the constraints imposed by the D, G, and Q conditions. However, it is not evident how to approach
⌬ji, for j ⫽ i, in terms of the occupation numbers.
Therefore, let’s rewrite the energy term in Eq. (43),
which involves ⌬ji as
冘⬘ ⌬ J ⫽ 冘 J * 冘⬘ ⌬ ,
ji ij
i
ij
i
j ⫽ i.
(54)
From Eq. (54), it is evident that g␣␣
ji ⱖ 0 if ⌬ji ⱖ
ni(nj ⫺ 1). This inequality is easy to satisfy on the
domain of allowed occupation numbers (nj ⱕ 1), if
we consider non-negative ⌬ji.
The opposite spin component consists entirely of
1 ⫻ 1 blocks G␣␤
ii,ii ⫽ 0, and 2 ⫻ 2 blocks
冉
G ␣␤
ij,ij
G ␣␤
ji,ij
冊
G ␣␤
ij, ji
.
G ␣␤
ji, ji
(55)
(58)
ji
j
where Ji* denotes the mean value of the Coulomb
interactions Jij for a given orbital i taking over all
orbitals j ⫽ i. From the property shown in Eq. (30)
follows immediately
冘⬘ ⌬ J ⫽ 冘 n 共1 ⫺ n 兲J*.
ji ij
1
共n ⫺ n j n i ⫹ ⌬ ji 兲,
2 i
(57)
4. A Practical Functional
i
ij
g ␣␣
ji ⫽
(56)
i
(59)
i
i
Inserting this expression into Eq. (43), one obtains
E⫽
冘 共2n h ⫹ n K 兲 ⫹ 冘⬘ 共2n n J ⫺ ⌳ K 兲
⫹ 冘 n 共1 ⫺ n 兲共K ⫺ 2J *兲.
2
i
i ii
ii
j
i
i ij
ji
ij
ij
i
i
ii
i
(60)
i
A further simplification of our NOF is accomplished by setting Ji* ⬇ Kii/2, which produces
E⫽
After some straightforward algebra, it can be
shown that blocks (55) afford the eigenvalues
INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY
冘 共2n h ⫹ n K 兲 ⫹ 冘⬘ 共2n n J ⫺ ⌳ K 兲.
i ii
i
2
i
ii
j
i ij
ji
ij
ij
(61)
1099
PIRIS
TABLE I ______________________________________________________________________________________________
Total energies (Etotal) in Hartrees.
Molecule
AlCl
AlF
AlH
BCl
Be2
BeH2
BeO
BeS
BF
BH
C2H2
CH4
Cl2
CO
CO2
CS
F2
FCl
FH
H2CO
H2N2
H2O
H2O2
HBO
HCF
HCl
HCN
HCP
HNO
HOF
HPO
Li2
LiCl
LiF
LiH
LIOH
Mg2
MgO
MgS
N2
N2O
Na2
NaCl
NaF
NaH
NaOH
NH3
O3
P2
PH3
1100
HFa
CCSD(T)b
NOFc
B3LYPd
⫺701.446543
⫺341.405383
⫺242.438008
⫺484.105019
⫺29.122716
⫺15.766705
⫺89.406757
⫺412.103398
⫺124.101178
⫺25.119105
⫺76.820612
⫺40.201379
⫺918.908639
⫺112.736756
⫺187.631787
⫺435.302218
⫺198.669746
⫺558.816103
⫺100.009834
⫺113.867947
⫺109.997791
⫺76.022615
⫺150.769507
⫺100.166126
⫺137.753632
⫺460.064368
⫺92.875178
⫺379.105181
⫺129.783338
⫺174.730223
⫺416.121769
⫺14.865878
⫺467.006944
⫺106.933139
⫺7.981141
⫺82.908772
⫺399.187445
⫺274.322470
⫺597.077741
⫺108.941801
⫺183.675227
⫺323.681228
⫺621.397562
⫺261.300190
⫺162.372474
⫺237.278603
⫺56.194962
⫺224.242821
⫺681.421021
⫺342.452229
⫺701.645139
⫺341.639667
⫺242.506581
⫺484.329332
⫺29.224189
⫺15.829649
⫺89.648394
⫺412.284575
⫺124.351814
⫺25.206100
⫺77.106349
⫺40.388310
⫺919.198123
⫺113.032821
⫺188.113590
⫺435.569554
⫺199.045358
⫺559.143280
⫺100.198698
⫺114.202886
⫺110.357170
⫺76.228954
⫺151.166824
⫺100.442361
⫺138.060398
⫺460.221565
⫺93.181241
⫺379.377798
⫺130.149630
⫺175.113429
⫺416.441974
⫺14.896032
⫺467.157908
⫺107.126788
⫺8.008224
⫺83.121379
⫺399.256740
⫺274.592547
⫺597.256828
⫺109.261984
⫺184.203202
⫺323.707095
⫺621.546221
⫺261.491762
⫺162.400444
⫺237.488190
⫺56.399981
⫺224.867269
⫺681.667624
⫺342.605974
⫺701.723456
⫺341.660582
⫺242.525697
⫺484.376709
⫺29.200175
⫺15.821505
⫺89.609227
⫺412.310091
⫺124.346169
⫺25.193976
⫺77.069774
⫺40.363928
⫺919.345443
⫺113.020010
⫺188.134537
⫺435.601712
⫺199.109071
⫺559.244622
⫺100.178202
⫺114.208902
⫺110.370685
⫺76.207940
⫺151.202528
⫺100.419037
⫺138.086920
⫺460.258382
⫺93.145714
⫺379.389201
⫺130.185528
⫺175.161571
⫺416.509129
⫺14.887910
⫺467.192191
⫺107.100782
⫺8.000625
⫺83.095776
⫺399.282202
⫺274.561446
⫺597.302717
⫺109.241787
⫺184.233374
⫺323.717745
⫺621.586299
⫺261.473972
⫺162.396537
⫺237.469800
⫺56.378970
⫺224.992524
⫺681.742405
⫺342.635555
⫺702.682030
⫺342.331111
⫺242.981911
⫺484.974255
⫺29.343773
⫺15.917811
⫺89.898440
⫺412.894462
⫺124.656132
⫺25.288104
⫺77.327715
⫺40.523216
⫺920.341830
⫺113.306694
⫺188.577339
⫺436.204876
⫺199.495477
⫺559.937606
⫺100.425817
⫺114.500848
⫺110.639297
⫺76.417892
⫺151.537846
⫺100.711017
⫺138.398508
⫺460.797390
⫺93.421806
⫺379.992926
⫺130.467334
⫺175.524723
⫺417.135327
⫺15.013967
⫺467.792715
⫺107.416662
⫺8.082268
⫺83.382938
⫺400.156464
⫺275.212883
⫺598.249807
⫺109.520563
⫺184.656028
⫺324.586846
⫺622.556987
⫺262.156146
⫺162.852762
⫺238.113220
⫺56.556343
⫺225.400708
⫺682.683866
⫺343.142663
(continued)
VOL. 106, NO. 5
TWO-ELECTRON CUMULANT IN NATURAL ORBITAL FUNCTIONAL THEORY
TABLE I ______________________________________________________________________________________________
(Continued)
HFa
CCSD(T)b
NOFc
B3LYPd
⫺395.122081
⫺398.673332
⫺290.001640
⫺291.229906
⫺363.776369
⫺686.438897
⫺547.165010
⫺395.425821
⫺398.832641
⫺290.109244
⫺291.366504
⫺364.055966
⫺686.663532
⫺547.686081
⫺395.444920
⫺398.871007
⫺290.137664
⫺291.394429
⫺364.079292
⫺686.742597
⫺547.805959
⫺396.057582
⫺399.388424
⫺290.613330
⫺291.886230
⫺364.716360
⫺687.690102
⫺548.579504
Molecule
PN
SH2
SiH2
SiH4
SiO
SiS
SO2
a
Hartree–Fock total energies.
CCSD(T) total energies.
c
Natural orbital functional total energies computed in this work.
d
B3LYP total energies.
b
Under these circumstances, the NOF (43) turns
out to be identical to the self-interaction-corrected
Hartree functional proposed by Goedecker and
Umrigar [6], except for the choice of phases given
by sign(⌳ji). The functional form (40) for the matrix
elements of ⌳ between HF occupied orbitals gives a
wrong description of the occupation numbers for
the lowest occupied levels. They are identically
equal to one. To ensure that these occupation numbers only are close to unity, we assume the form
(see, e.g., Ref. [5]):
⌳ ji ⫽
⌫ ␯␩ ⫽
j
J ␩␯
⫽
i ⫽ j.
冘 C ␸ 共r兲.
(63)
␯i ␯
The electronic energy (61) will then be a functional
冘 再冋2h ⌫ ⫹ 冘 n K ⌫ 册
␯␩
2
i
i
⫹
冘⬘ [2n n J
j
j i ␩␯
ij
i
⌫ ␯␩
⫽ C ␯iC ␩i
(65)
j
␭␮
(66)
␭␮
j
K ␩␯
⫽
冘 具␩␮兩␭␯典⌫
j
␭⌫
.
(67)
␭␮
The orthonormality condition (4) reads as follows:
冘C S
␮␯
C ␯ i ⫽ ␦ ji ,
(68)
(62)
␯
␩␯ ␯␩
,
冘 具␩␮兩␯␭典⌫
␮j
Let us apply the well-known procedure of taking
molecular orbitals as linear combination of atomic
orbitals (MO-LCAO),
E关兵C i其, 兵ni其兴 ⫽
i
␯␩
␯␮
i ⱕ nco, j ⱕ nco,
␺ i共r兲 ⫽
i
i
冑 n jn i ⫹ 冑 共1 ⫺ n j兲共1 ⫺ n i兲
if
冘 n⌫
i
i
␩␯ ␯␩
冎
j
i
⫺ ⌳jiK␩␯
]⌫␯␩
,
(64)
where it has been introduced the following matrices:
where S␮␯ ⫽ 具␸␮兩␸␯典 is the overlap matrix.
5. Methodological Details
Direct minimization of the energy functional of
Eq. (64) is required, subject to the following constraints: (i) the N-representability condition of the
one-matrix (0 ⱕ ni ⱕ 1); (ii) the constant number of
particles [Eq. (21)]; and (iii) the orthonormality condition [Eq. (68)].
One has to calculate the gradient of the functional both with respect to natural orbital coefficients {Ci} and the occupation numbers {ni}. Since
the minimization with respect to occupations is
much less expensive than with respect to the orbitals, one can decouple the variation of the occupation numbers from that of the natural orbitals, a
procedure used by us in the improved Bardeen–
Cooper–Schrieffer (IBCS) method [20]. In the inner
INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY
1101
PIRIS
loop, we find the optimal occupation numbers for a
given set of orbitals under constraint 2. Bounds on
the occupation numbers are enforced by setting
ni ⫽ (sin ␥i)2 and varying the ␥i without constraints.
In the outer loop, we minimize with respect to the
orbital coefficients under the constraints 3 of mutual orthonormality. Both the inner- and outer-loop
optimizations have been implemented using a sequential quadratic programming (SQP) method
[23].
6. Results
In this section, calculations of total energies and
dipole moments for selected molecules using contracted Gaussian basis sets 6-31G** [24] are presented. Computationally speaking, NOF theory in
its current form is very demanding. Therefore, we
have chosen a medium-size basis set for the calculations, and we have compared the results with
those using other methods at the same level. We are
aware of the fact that very large basis sets are
required to estimate experimental values.
Among the approaches compared are the coupled cluster technique, including all single and
double excitations and a perturbational estimate of
the connected triple excitations [CCSD(T)], as well
as the Becke-3–Lee–Yang–Parr (B3LYP) density
functional [25]. The CCSD(T) and B3LYP values
were calculated with the Gaussian 94 system of
programs [26], using the basis set keyword 5D.
Table I presents the values obtained for the total
energies of 57 molecules, employing the experimental geometry [27, 28]. For comparison, Table I includes the total energies calculated at the CCSD(T)
and B3LYP levels. According to Table I, the values
we have obtained are more like CCSD(T) calculations, which are very accurate results for the basisset correlation energies of these small molecules.
The B3LYP values, as is well known, tend to be too
low. We note that the percentage of the correlation
energy obtained by CCSD(T) decreases as the number of electrons increases, whereas our functional
keeps giving a slightly larger portion of the correlation energy (e.g., AlCl, SiS, P2, SO2, Cl2).
For molecules with dipole moments (␮) different
from zero, we have also evaluated this property
(Table II). For comparison, Table II includes the
available experimental data [29] of ␮ and those
calculated with the Gaussian 94 system of programs
at the CCSD(T) and B3LYP levels.
1102
TABLE II ______________________________________
Dipole moments (␮) in Debyes.
Molecule
HFa
CCSD(T)b
B3LYPc
NOFd
Expe
AlCl
AlF
AlH
BCl
BeO
BeS
BF
BH
CO
CS
FCl
FH
H2CO
H2O
H2O2
HBO
HCF
HCl
HCN
HCP
HNO
HOF
HPO
LiCl
LiF
LiH
LIOH
MgO
MgS
N2O
NaCl
NaF
NaH
NaOH
NH3
O3
PH3
PN
SH2
SiH2
SiO
SiS
SO2
1.72
1.06
0.49
0.94
6.91
6.42
0.87
1.48
0.33f
1.26
1.26
1.98
2.75
2.20
1.86
3.11
1.45
1.48
3.23
0.71
2.02
2.24
2.93
7.43
6.20
5.89
4.27
7.81
9.03
0.60f
9.39
8.00
6.84
6.33
1.89
0.78
0.80
2.91
1.38
0.49
3.36
2.51
2.21
1.62
1.12
0.15
1.00
5.33
4.59
0.88
1.03
0.07
1.76
1.02
1.87
2.18
2.09
1.75
2.38
1.23
1.37
2.88
0.65
1.63
1.98
2.24
7.05
5.86
5.58
3.85
5.14
6.38
0.07f
8.97
7.52
6.19
5.85
1.81
0.49
0.80
2.46
1.30
0.31
2.59
1.53
1.78
1.57
0.99
0.28
1.21
5.58
5.14
1.10
1.32
0.10
1.57
0.91
1.82
2.17
2.04
1.72
2.45
1.34
1.43
2.88
0.71
1.61
1.96
2.01
6.91
5.62
5.58
3.65
6.36
7.12
0.01
8.61
7.05
5.96
5.48
1.79
0.62
0.96
2.45
1.41
0.52
2.60
1.74
1.68
1.60
1.09
0.23
1.00
6.16
5.38
0.96
1.28
0.07f
1.47
0.75
1.84
2.36
2.08
1.65
2.70
1.22
1.30
3.01
0.59
1.61
1.85
2.33
7.21
6.01
5.75
4.17
5.21
6.68
0.03
9.03
7.82
6.44
6.11
1.76
0.50
0.60
2.59
1.19
0.23
2.81
1.84
1.56
—
1.53
—
—
—
—
—
—
0.11
1.98
0.88
1.82
2.33
1.85
2.20
—
—
1.08
2.98
0.39
1.67
2.23
—
7.13
6.33
5.88
4.75
—
—
0.17
9.0
8.16
—
—
1.47
0.53
0.58
2.75
0.97
—
3.10
1.73
1.63
a
Hartree–Fock dipole moments.
CCSD(T) dipole moments.
c
B3LYP dipole moments.
d
Natural orbital functional dipole moments computed in this
work.
e
Experimental dipole moments from Ref. [29].
f
This value has an opposite sign relative to the experimental
value.
b
VOL. 106, NO. 5
TWO-ELECTRON CUMULANT IN NATURAL ORBITAL FUNCTIONAL THEORY
The dipole moments obtained with the correlated methods are in good agreement with the experimental data considering the basis sets (6-31G**)
used for these calculations. For the reported molecules, the correlated dipole moments are lower
compared with HF ␮, except for BCl, BF, and CS.
For PH3 and SiH2, we have to mention that B3LYP
dipole moments are increased with respect to the
HF result, whereas the NOF and CCSD(T) values
continue to be smaller than it. On the contrary, in
the case of the AlF molecule, the B3LYP method
decreases ␮ with respect to the HF value, whereas
the NOF and CCSD(T) yield a higher dipole moment. Important cases are the CO and N2O molecules for which the HF approximation gives a dipole moment in the wrong direction, whereas
correlation methods approach it to the experimental value.
The quality of natural orbitals is critical to the
accuracy of NOF theory. For example, in the case of
the CO molecule, we can achieve the sign inversion
of the dipole moment (␮ ⫽ 0.03 Debyes) by adding
only one basis function (6-311G**). It is expected
that better agreement can be obtained with further
improvement of the basis sets.
By considering the mean value theorem and the
partial sum rule for matrix ⌬, we achieve a practical
functional that is close to the self-interaction-corrected GU functional. Despite the previously reported N-representability violations of the GU
functional, the cumulant expansion examined in
the present work can correct these positivity problems, at the same time satisfying the trace relation
of the two-matrix. In contrast, our ansatz (62) reproduces properly the occupation numbers for
lower occupied levels. An improvement in our approach requires better approximations for the mean
values { Ji*} of the Coulomb interactions.
A representative set of 57 molecules is investigated. Comparison with other theoretical methods
shows that the presented NOF provides total energies closer to accurate ab initio methods than to
DFT energies. The agreement between theory and
experiment for ␮ is satisfactory, considering the
basis sets used. The accuracy of the dipole moments
in all calculational approaches in the present study
appears to be comparable to each other. In conclusion, the encouraging results obtained in this work
demonstrate that our NOF can be used to predict
other properties.
References
7. Concluding Remarks
Making use of the known cumulant expansion, a
new reconstructive functional for the two-matrix is
proposed. Its explicit antisymmetric form leads to a
better description of the pair density for parallelspin electrons with respect to our previous proposal. The dependence obtained in the IBCS
method was also considered for the opposite-spin
component of the cumulant.
The functional given in Eq. (35) can be reduced
to the exact energy expression for singlet ground
states of two-electron closed-shell systems such as
H2 or He. This permits us to generalize its functional form for N-electron systems, except for the
off-diagonal elements of a symmetric matrix ⌬.
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of the two-matrix. To this end, the well-known
necessary D, G, and Q conditions are discussed.
The analytic determined eigenvalues provide rigorous bounds on the magnitudes {⌬ji} to guarantee
that our reconstructed functional satisfies these
positivity conditions.
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